X_i = f(L,K,M,R)

where X_i is the quantity produced of a particular good or service and:

• L represents the quantity and ability of labor input available to the production process.
• K represents capital input, machinery, transportation equipment, and other types of intermediate goods.
• M represents land, natural resources and raw material inputs for production, and
• R represents entrepreneurship, organization and risk-taking.

A positive relationship exists among these inputs and the output such that greater availability of any of these factors will lead to a greater potential for producing output. In addition, all factors are assumed to be essential for production to take place. The functional relationship f(.) represents a certain level of technology and know how, that presently exists, for conversion these inputs into output such that any technological improvements can also lead to the production of greater levels of output.

X_i = f(L;K,M,R)

X_i = f(L)

For example we could develop a short run model for agricultural production where the output is measures as kilograms of grain and labor is the variable input. The fixed factors of production include the following:

• 1 plow
• 1 tractor.... capital
• 1 truck
• 1 acre of land
• 10 kilograms of seed grain

In this example we find that each time we add one more unit of labor, output increases by 100 kg. The third column MP_L defines this relationship. This column measures the marginal productivity of labor -- a measure of the contribution of each additional unit of labor input to the level of output. In this case, we have a situation of constant marginal productivity which is unrealistic with production in the short run. Constant marginal productivity implies that as labor input increases, output always increases without bound -- a situation difficult to imagine with limited capital and one acre of land.

A more realistic situation would be that of diminishing marginal productivity where increasing quantities of a single input lead to less and less additional output. This property is just an acknowledgment that it is impossible to produce an infinite level of output when some factors of production (machines or land) fixed in quantity. Numerically, we can model diminishing marginal productivity as follows:

In this case, additional labor input results in additional output. However, the contribution of each additional unit of labor is less than previous units such that the sixth unit of labor contributes nothing to output. With 5 or 6 workers, the available amount of land cannot support additional output.

A short run production relationship can be modeled in the diagram below. In this example, labor is the variable factor input and land, capital, and entrepreneurship are fixed in quantity. There is a positive relationship between labor input and output levels, however, as additional labor in used, less and less additional output is produced (click on the second button). The shape of this production function is consistent with the law of diminishing marginal productivity.

In the table below, we can model this competition for resources between two goods: Apples and Bread. In this example, an additional unit of labor directed to bread production allows for producing 25 additional units of bread (the marginal productivity of labor in bread production is constant). Separately, additional units of labor applied to apple production allows for the producing anywhere from 0 to 185 units of apples.

Working with an assumption that the amount of labor input is fixed in supply at 7 units, this labor must be shared between bread and apple production. If we happen to be producing 185 units of apples (point A), then all 7 units of labor are being used for this purpose. The desire to produce bread requires a reallocation of labor from apple production to bread production. In the table above, we can show this as a movement from point A to point B. We are able to produce 25 units of bread (0-25 units) but at the expense producing 6 apples (185-179 units). These six apples represent the opportunity cost of bread production (i.e., each unit of bread "costs" 6/25 unit of apples. If we move from point B to point C, the additional 25 units of bread (25-50), come at the expense of 16 apples (179-163 units). Because of diminishing marginal productivity in apple production, the production of 25 additional units of bread requires that more and more apples are given up. Stated differently, we can say that the opportunity cost of producing bread is increasing.

The diagram below summarizes the numbers in the above table. Points on the blue curve -- the Production Possibilities Frontier represent an efficient use of resources. Points within the curve represent inefficient production levels -- resources and technology allow for producing more of good X, good Y, or more of both. Movement along the curve (use the scrollbar to see changes) imply that a tradeoff exists in production when resources are scarce or fixed in supply. Finally points (combinations of the two goods) beyond the frontier are unattainable with existing levels of technology and resource availability.

MRT = MP_y /MP_x = Marginal Cost_x /Marginal Cost_y

given that: Marginal Cost_i = wage rate /MP_i

If resources were to be allocated in the opposite direction, the same would be true -- the opportunity cost of producing more of good Y would also increase in terms of foregone production of good X.

P^R = [P_x / P_Y] = [P_bread / P_apples] = 1.50.

If we compare the relative price of bread to the opportunity cost of bread, we find that the value of bread in terms of apples is greater than the opportunity cost of producing bread:

one loaf of bread is worth 1.50 apples
one loaf of bread costs 1.20 apples, or

P_bread / P_apples > MRT

P^R = MRT

P_bread / P_apples = MC_bread / MC_apples

If the price of bread were to fall, say to $2.00 per unit (click on the Next button), then the relative price of bread would be equal to 1.0 (i.e., each unit of bread is exactly equal to the value of one unit of apples) and P^R . With this change, the value of bread, in terms of apples, is less than the opportunity costs of producing bread. In this case we should allocate resources away from bread production (click on the Next button again). This reallocation will cause the opportunity cost of bread to fall until it is just equal to the prevailing relative price. (Click on the Next button again to repeat.)

This example above, demonstrates how the relationship between relative prices (acting as signals about the "value" place on products by consumers) and opportunity costs (that represent the underlying characteristics of production technology) determine an efficient use of resources. As relative prices change, resources will be reallocated to that good that is valued more highly taking into account the opportunity costs of additional production of that good.

• Diminishing Marginal Productivity
• Inefficient Production
• Long Run Production
• Marginal Productivity of Labor
• Marginal Rate of Transformation
• Opportunity Cost
• Production Function
• Production Possibilities
• Relative Prices
• Short Run Production
• Technology
• Unattainable Output Combinations